Self-tuning for disturbance transmission decoupling in active vehicle suspensions

With active vehicle suspension, one can tailor a vehicles response to load and inertial disturbances without affecting the vehicle response to road disturbances. This decoupling is achieved in [1] and [2] using a filtered combination of measured
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  Proceedings in Applied Mathematics and Mechanics, 29 May 2008 Adaptive disturbance transmission decoupling in active vehicle suspen-sions Mark Readman ∗ 1 , Martin Corless 1,2 , Carlos Villegas 1 ,  and  Robert Shorten 1 1 Hamilton Institute,National University of Ireland, Maynooth, Co. Kildare, Ireland 2 School of Aeronautics & Astronautics, Purdue University-West Lafayette, IN, USAWith active vehicle suspension, one can tailor a vehicles response to load and inertial without affecting the vehicle response toroad disturbances. This decoupling is achieved in [1] and [2] using a filtered combination of measured signals. These filtersrequire exact knowledge of certain vehicle parameters including vehicle mass to achieve the desired decoupling. Here wepropose a parameter adaptive version of these filters which does not require knowledge of vehicle parameters. Copyright line will be provided by the publisher 1 Introduction In [1] and [2] an active suspension control strategy is presented. that uses combined filtered sensor measurements so that the ride height controller is not affected by road disturbance inputs. We refer to this arrangementas the Williams filter. In its mostbasic form, the Williams filter, for an active suspension, gives a method of specifying the transmission from road disturbanceto chassis vertical acceleration, while at the same time responding to load disturbances from the car body. Williams filterassumes perfect knowledgeof some suspension and vehicle parameters, some of which may be time varying. In this paper, weshow how one may readily obtain an adaptive version of the Williams filter which does not require knowledge of the vehicleparameters and which can accommodate changes in vehicle parameters. We also consider a specific adaption scheme anddemonstrate its stability using Lyapunov type arguments. 2 Decoupling control structure The quarter car model can be described by m s ¨ z s  =  − c s (˙ z s −  ˙ z u ) − k s ( z s − z u − u ) +  F  s  (1) m u ¨ z u  =  c s (˙ z s −  ˙ z u ) +  k s ( z s − z u − u ) − k t ( z u − z r )  (2)The control input  u  is the deflection of the active suspension system which is in series with the suspension spring of stiffness k s . We assume, as in [1], that  ¨ z s  and  z s − z u  can be measured. We wish to design a controller for  u  to achieve some desirableresponse of the vehicle to the disturbance  F  s  without affecting the vehicle response to  z r  [4]. To achieve this, we first obtaina signal which is independent of   z r  and use this as input to the “ride height controller” which generates  u . If, as in [2], weconsider  F  ( s ) = 1 / ( c s s  +  k s )  and letting  α  =  k s /c s , β   =  m s /c s  then e  =  W  ( s )¨ z s  +  z s − z u  where  W  ( s ) =  β s  +  α  (3) 3 The Adaptive Williams filter The filters proposed in [1] and [2] can never be implemented due to uncertainty in the suspension and vehicle parameters. Therefore road noise will not be cancelled exactly at the controller. Since the filter  W  ( s )  in (3) only contains two parameters, α  and  β  , one approach is to replace  α  and  β   in (3) with some adaptively tuned estimates  ˆ α  and  ˆ β  . This results in the adaptiveWilliams filter: To obtain adaptation laws for the estimated parameters, we note that equation (1) can be written as ˙ η  +  α ( η − u ) +  β  ¨ z s  +  γ   =  w  (4)where  η  :=  z s − z u . If   w  is a low frequency signal, we can mitigate its effect by applying a high pass filter  T   to the abovedifferential equation. Comment :  We now note that (4) describes a simple linear first order system which depends linearly on the constantparameters  α ,  β   and  γ  ; also all the signals  η ,  u  and  ¨ z s  are available. Thus, we can use some standard parameter estimationschemes to obtain estimates  ˆ α ,  ˆ β   and  ˆ γ   of   α ,  β   and  γ  . We use a simple parameter estimation scheme which is based on thoseused in the adaptive control literature; see, for example [3]. To obtain convergence of   (ˆ α,  ˆ β  )  to  ( α,β  )  one needs the signals η − u  and  ¨ z s  to be ”sufficiently rich”. ∗ Corresponding author: e-mail: Copyright line will be provided by the publisher  PAMM header will be provided by the publisher 2 0 5 10 15−2024Parameter Estimates      α 0 5 10 1500.050.1        β 0 5 10 15−0.0100.010.02      γ Time (Sec) Fig. 1  Parameter estimates 0 5 10 15−1−0.8−0.6−0.4− 10 −3 Filter Output       e        (        t        ) Time(Sec) Fig. 2  Filter output  e 4 Simulation Results Fornumericalsimulationresults, we used  m s  = 250kg ,  c s  = 4kNs / m  and k s  = 12kN / m from[2]; this results in  α  = 0 . 0625 and  β   = 3 . The ride height controller is chosen to be the robust loop shaping controller also obtained from [2]. The initialsimulation results are shown in Figure 1. The parameters converge to their correct values after approximately 5 seconds andthe effect of road noise is eliminated from the signal  e . At 5 seconds a 100N step change in the down force is applied and at10 seconds the sprung mass  m s  is increased by  25% . The parameter estimates then converge to the new correct values androad noise is eliminated from  e . 5 CONCLUSIONS Thedecouplingcontrolschemeadaptivelycancels theeffect ofroadnoise seenby the rideheightcontroller. Simulationresultsdemonstrate the effectiveness of the adaptive decoupling controller. Acknowledgements  This work was supported by SFI Investigator award 04/INI/I478 and the EU Strep CeMACS. References [1] R. A. Williams and A. Best. Control of low frequency active suspension.  International Conference on Control’94 , pp. 338–343, 1994.[2] M. C. Smith and F.-C. Wang, Controller parameterization for disturbance response decoupling: Application to vehicle active suspen-sion control,  IEEE Transactions on Control Systems Technology , Vol. 10, pp. 393-407, 2002.[3] K. B Narendra and A. M. Annaswamy,  Stable Adaptive Systems , Prentice Hall, 1989.[4] S. T¨urkay and H. Akc¸ay, A study of random vibration characteristics of the quarter-car model,  Journal of Sound and Vibration , Vol.282, pp. 111-124, 2005. Copyright line will be provided by the publisher
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